Factoring Banded Permutations and Bounds on the Density of Vertex Identifying Codes on the Infinite Snub Hexagonal Grid

A permutation may be characterized as b-banded when it moves no element more than b places. Every permutation may be factored into 1-banded permutations. We prove that an upper bound on the number of tridiagonal factors necessary is 2b-1, verifying a conjecture of Gilbert Strang. A vertex identifying code of a graph is a subset D of the graph's vertices with the property that for every pair of vertices v1 and v2, N[v1]∩D and N[v2]∩D are distinct and nonempty, where N[v] is the set of all vertices adjacent to v, including v. We compute an upper bound of 1/3 and a strict lower bound of 3/10 for the minimum density of a vertex identifying code on the infinite snub hexagonal grid.